We consider a flip dynamics for directed (1+d)-dimensional lattice paths with length L. The model can be interpreted as a higher dimensional version of the simple exclusion process, the latter corresponding to the case d=1. We prove that the mixing time of the associated Markov chain scales like L^2\log L up to a d-dependent multiplicative constant. The key step in the proof of the upper bound is to show that the system satisfies a logarithmic Sobolev inequality on the diffusive scale L^2 for every fixed d, which we achieve by a suitable induction over the dimension together with an estimate for adjacent transpositions. The lower bound is obtained with a version of Wilson's argument for the one-dimensional case.